The local semicircle law for a general class of random matrices

Laszlo Erdos Institute of Science and Technology Austria Antti Knowles New York University Horng-Tzer Yau Harvard University Jun Yin University of Wisconsin-Madison

Probability mathscidoc:1608.28017

Electronic Journal of Probability, 18, (12), 2140-2154, 2013
We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [14] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \E \abs{h_{ij}}^2$. As a consequence, we prove the universality of the local $n$-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width $W\gg N^{1-\e_n}$ with some $\e_n>0$ and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [17,19,6].
Random band matrix; local semicircle law; universality; eigenvalue rigidity
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@inproceedings{laszlo2013the,
  title={The local semicircle law for a general class of random matrices},
  author={Laszlo Erdos, Antti Knowles, Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823164502007757414},
  booktitle={Electronic Journal of Probability},
  volume={18},
  number={12},
  pages={2140-2154},
  year={2013},
}
Laszlo Erdos, Antti Knowles, Horng-Tzer Yau, and Jun Yin. The local semicircle law for a general class of random matrices. 2013. Vol. 18. In Electronic Journal of Probability. pp.2140-2154. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823164502007757414.
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